A simple field theory approach is developed to model the properties of charged, dielectric bodies and their associated counterions. This predictive theory is able to accurately describe the properties of systems (as compared to computer simulation data) from the weak coupling limit, where the Poisson-Boltzmann theory works well, through to the strong coupling limit. In particular, it is able to quantitatively describe the attraction between like-charged plates and the influence of image charge interactions.
A simple model is presented for the appearance of attraction between two like charged polyions inside a polyelectrolyte solution. The polyions are modeled as rigid cylinders in a continuum dielectric solvent. The strong electrostatic interactions between the polyions and the counterions results in counterion condensation. If the two polyions are sufficiently close to each other their layers of condensed counterions can become correlated resulting in attraction between the macromolecules. To explore the counterion induced attraction we calculate the correlation functions for the condensed count…
The counterion-condensation transition at charged cylinders is studied using Monte-Carlo simulation methods. Employing logarithmically rescaled radial coordinates, large system sizes are tractable and the critical behavior is determined by a combined finite-size and finite-ion-number analysis. Critical counterion localization exponents are introduced and found to be in accord with mean-field theory both in 2 and 3 dimensions. In 3D the heat capacity shows a universal jump at the transition, while in 2D, it consists of discrete peaks where single counterions successively condense.
We evaluate exactly the statistical integral for an inhomogeneous one-dimensional counterion-only Coulomb gas between two charged boundaries and from this compute the effective interaction, or disjoining pressure, between the bounding surfaces. Our exact results are compared with the limiting cases of weak and strong coupling which are the same for 1D and 3D systems. For systems with a large number of counterions it is found that the weak coupling (mean-field) approximation for the disjoining pressure works perfectly and that fluctuations around the mean-field in 1D are much smaller than in 3D…
We investigate the statistical mechanics of an inhomogeneous Coulomb fluid composed of charged particles with static polarizability. We derive the weak- and the strong-coupling approximations and evaluate the partition function in a planar dielectric slab geometry with charged boundaries. We investigate the density profiles and the disjoining pressure for both approximations. Comparison to the case of non-polarizable counterions shows that polarizability brings important differences in the counterion density distribution as well as the counterion mediated electrostatic interactions between cha…
By a combination of Monte Carlo simulations and analytical calculations, we investigate the effective interactions between highly charged planar interfaces, neutralized by mobile counterions (salt-free system). While most previous analysis have focused on point-like counterions, we treat them as charged hard spheres. We thus work out the fate of like-charge attraction when steric effects are at work. The analytical approach partitions counterions in two sub-populations, one for each plate, and integrates out one sub-population to derive an effective Hamiltonian for the remaining one. The effec…
We examine the effective counterion-mediated electrostatic interaction between two like-charged dielectric cylinders immersed in a continuous dielectric medium containing neutralizing mobile counterions. We focus on the effects of image charges induced as a result of the dielectric mismatch between the cylindrical cores and the surrounding dielectric medium and investigate the counterion-mediated electrostatic interaction between the cylinders in both limits of weak and strong electrostatic couplings (corresponding, e.g., to systems with monovalent and multivalent counterions, respectively). T…
We address the critical and universal aspects of counterion-condensation transition at a single charged cylinder in both two and three spatial dimensions using numerical and analytical methods. By introducing a novel Monte-Carlo sampling method in logarithmic radial scale, we are able to numerically simulate the critical limit of infinite system size (corresponding to infinite-dilution limit) within tractable equilibration times. The critical exponents are determined for the inverse moments of the counterionic density profile (which play the role of the order parameters and represent the inver…
We present a simple model for the possible mechanism of appearance of attraction between like charged polyions inside a polyelectrolyte solution. The attraction is found to be short ranged, and exists only in presence of multivalent counterions. The attraction is produced by the correlations in the condensed layers of counterions surrounding each polyion, and appears only if the number of condensed counterions exceeds the threshold, $ n > Z/2 α$, where $α$ is the valence of counterions and $Z$ is the polyion charge.
We simulate model systems of charged spherical particles in their counterion solution and measure the thermodynamic pressure and the pair distribution function from which we derive effective potentials of mean force. For a system with only electrostatic and hard core interactions, we investigate the effective potential between two like-charged spheres in divalent counterion solution as a function of concentration. We find a strong attractive interaction for high concentration and a global repulsive effective interaction for dilute systems. The results indicate a first order phase transition in…
The persistence length of a single, strongly charged, stiff polyelectrolyte chain is investigated theoretically. Path integral formulation is used to obtain the effective electrostatic interaction between the monomers. We find significant deviations from the classical Odijk, Skolnick and Fixman (OSF) result. An induced attraction between monomers is due to thermal fluctuations and correlations between bound counterions. The electrostatic persistence length is found to be smaller than the OSF value and indicates a possible mechanical instability (collapse) for highly charged polyelectrolytes wi…
We investigate the phenomenon of counterion condensation in a solution of highly charged rigid polyelectrolytes within the cell model. A method is proposed which -- based on the charge distribution function -- identifies both the fraction of condensed ions and the radial extension of the condensed layer. Within salt-free Poisson-Boltzmann (PB) theory it reproduces the well known fraction 1-1/xi of condensed ions for a Manning parameter xi>1. Furthermore, it predicts a weak salt dependence of this fraction and a breakdown of the concept of counterion condensation in the high salt limit. We comp…
Realistic charged macromolecules are characterized by discrete (rather than homogeneous) charge distributions. We investigate the effects of surface charge discretization on the counterion distribution at the level of mean-field theory using a two-state model. Both planar and cylindrical geometries are considered; for the latter case, we compare our results to numerical solutions of the full Poisson-Boltzmann equation. We find that the discretization of the surface charge can cause enhanced localization of the counterions near the surface; for charged cylinders, counterion condensation can exc…
Within a mean field treatment of the interaction between two oppositely charged plates in a salt free solution, the distance at which a transition from an attractive to a repulsive regime appears can be computed analytically. The mean field description however breaks down under strong coulombic couplings, that can be achieved at room temperature with multivalent counter-ions and highly charged surfaces. Making use of the contact theorem and simple physical arguments, we propose explicit expressions for the equation of state in several situations at short distances. The possibility of Bjerrum p…
Using a generalization of the Poisson-Boltzmann equation, dynamics of counterion condensation is studied. For a single charged plate in the presence of counterions, it is shown that the approach to equilibrium is diffusive. In the far from equilibrium case of a moving charged plate, a dynamical counterion condensation transition occurs at a critical velocity. The complex dynamic behavior of the counterion cloud is shown to lead to a novel nonlinear force-velocity relation for the moving plate.
Screening of a macroion such as a charged solid particle, a charged membrane, double helix DNA or actin by multivalent counterions is considered. Small colloidal particles, charged micelles, short or long polyelectrolytes can play the role of multivalent counterions. Due to strong lateral repulsion at the surface of macroion such multivalent counterions form a strongly correlated liquid, with the short range order resembling that of a Wigner crystal. These correlations create additional binding of multivalent counterions to the macroion surface with binding energy larger than $k_BT$. As a resu…
We report a mechanism which can lead to long range attractions between like-charged spherical macroions, stemming from the existence of metastable ionized states. We show that the ground state of a single highly charged colloid plus a few excess counterions is overcharged. For the case of two highly charged macroions in their neutralizing divalent counterion solution we demonstrate that, in the regime of strong Coulomb coupling, the counterion clouds are very likely to be unevenly distributed, leading to one overcharged and one undercharged macroion. This long-living metastable configuration i…
We study a 2D system of identical mobile particles on the surface of a cylinder of finite length $d$ and circumference $W$, immersed in a medium of dielectric constant $\varepsilon$. The two end-circles of the cylinder are like-charged with the fixed uniform charge densities, the particles of opposite charge $-e$ ($e$ being the elementary charge) are coined as ``counterions''; the system as a whole is electroneutral. Such a geometry is well defined also for finite numbers of counterions $N$. Our task is to derive an effective interaction between the end-circles mediated by the counterions in t…
Charged soft-matter systems--such as colloidal dispersions and charged polymers--are dominated by attractive forces between constituent like-charged particles when neutralizing counterions of high charge valency are introduced. Such counter-intuitive effects indicate strong electrostatic coupling between like-charged particles, which essentially results from electrostatic correlations among counterions residing near particle surfaces. In this paper, the attraction mechanism and the structure of counterionic correlations are discussed in the limit of strong coupling based on recent numerical an…
We reformulate the theory of strong electrostatic coupling in order to describe an asymmetric electrolyte solution of monovalent salt ions and polyvalent counterions using field-theoretical techniques and Monte-Carlo simulations. The theory is based on an asymmetric treatment of the different components of the electrolyte solution. The weak coupling Debye-Huckel approach is used in order to describe the monovalent salt ions while a strong coupling approach is used to tackle the polyvalent counterions. This combined weak-strong coupling approach effectively leads to dressed interactions between…
Strong coupling phenomena, such as the like charged macroions attraction, opposite charged macroions repulsion, charge renormalization or charge inversion, are known to be mediated by multivalent counterions. Most theories treat the counterions as point charges, and describe the system by a single coupling parameter that measures the strength of the Coulomb interactions. In many biological systems, the counterions are highly charged and have finite sizes and can be well-described by polyelectrolytes. The shapes and orientations of these polymer counterions play a major role in the thermodynami…
Recent simulation results imply the lowering of the ground-state correlation energy per counterion at a charged planar wall, compared with that of the 2D and 3D one-component plasma systems. Our aim is to correctly evaluate the ground-state energy of strongly-coupled counterion systems by considering a quasi-2D bound state where bound counterions are confined to a layer of molecular thickness. We use a variational approach based on the Gibbs-Bogoliubov inequality for the lower-bound free energy so that the liquid-state theory can be incorporated into the formulations. The soft mean spherical a…
Polyamidoamine (PAMAM) dendrimers, being protonated under physiological conditions, represent a promising class of nonviral, nano-sized vectors for drug and gene delivery. We performed extensive molecular dynamics simulations of a generic model dendrimer in a salt-free solution with dendrimer's terminal beads positively charged. Solvent molecules as well as counterions were explicitly included as interacting beads. We find that the size of the charged dendrimer depends non-monotonically on the strength of electrostatic interactions demonstrating a maximum when the Bjerrum length equals the dia…
Bending rigidity of a charged membrane or a charged polyelectrolyte screened by monovalent counterions is known to be enhanced by electrostatic effects. We show that in the case of screening by multivalent counterions the electrostatic effects reduce the bending rigidity. This inversion of the sign of the electrostatic contribution is related to the formation of two-dimensional strongly correlated liquids (SCL) of counterions at the charged surface due to strong lateral repulsion between them. When a membrane or a polyelectrolyte is bent, SCL is compressed on one side and stretched on the othe…
We study the spatial and orientational distribution of rodlike counterions (such as mobile nanorods) as well as the effective interaction mediated by them between two plane-parallel surfaces that carry fixed (quenched) heterogeneous charge distributions. The rodlike counterions are assumed to have an internal charge distribution, specified by a multivalent monopolar moment and a finite quadrupolar moment, and the quenched surface charge is assumed to be randomly distributed with equal mean and variance on the two surfaces. While equally charged surfaces are known to repel within the traditiona…
We investigate the ion distribution and overcharging at charged interfaces with dielectric inhomogeneities in the presence of asymmetric electrolytes containing polyvalent and monovalent ions. We formulate an effective "dressed counterion" approach by integrating out the monovalent salt degrees of freedom and show that it agrees with results of explicit Monte-Carlo simulations. We then apply the dressed counterion approach within the framework of the strong-coupling theory, valid for polyvalent ions at low concentrations, which enables an analytical description for salt effects as well as diel…
We compare the recent experimentally measured forces between charged colloidal particles, as well as their effective surface potentials (surface charge) in the presence of multivalent counterions in a bathing monovalent salt solution, with the predictions of the dressed ion theory of strongly charged colloidal systems. The benchmark for comparison is provided by the DLVO theory and the deviations from its predictions at small separations are taken as an indication of the additional non-DLVO attractions that can be fitted by an additional phenomenological exponential term. The parameters charac…
Adsorption of multivalent counterions on the charged surface of a macroion is known to lead to inversion of the macroion charge due to the strong lateral correlations of counterions. We consider a nontrivial role of the excluded volume of counterions on this effect. It is shown analytically that when the bare charge of macroion increases, its net charge including the adsorbed counterions oscillates with the number of their layers. Charge inversion vanishes every time the top layer of counterions is completely full and becomes incompressible. These oscillations of charge inversion are confirmed…
We predict theoretically that long-wavelength surface charge modulations universally reduce the pressure between the charged surfaces with counterions compared with the case of uniformly charged surfaces with the same average surface charge density. The physical origin of this effect is the fact that surface charge modulations always lead to enhanced counterion localization near the surfaces, and hence, fewer charges at the midplane. We confirm the last prediction with Monte Carlo simulations.
We present general arguments for the importance, or lack thereof, of the structure in the charge distribution of counterions for counterion-mediated interactions between bounding symmetrically charged surfaces. We show that on the mean field or weak coupling level, the charge quadrupole contributes the lowest order modification to the contact value theorem and thus to the intersurface electrostatic interactions. The image effects are non-existent on the mean-field level even with multipoles. On the strong coupling level the quadrupoles and higher order multipoles contribute additional terms to…
The osmotic coefficient of B-DNA in water may, in moderately dilute solutions, deviate as much as 100 % from predictions based on a simple 'counterion condensation' theory. We determine the results for osmotic properties via a cell model description of the ionic atmosphere near a cylindrical polyelectrolyte. The cell model predictions for the osmotic properties disagree with predictions based on simple condensation theory, but are in surprisingly good harmony with experimental findings. We argue that the neglect of finite-radius effects makes simple condensation theory inapplicable at all but …
We study thermal equilibrium of classical pointlike counterions confined between symmetrically charged walls at distance $d$. At very large couplings when the counterion system is in its crystal phase, a harmonic expansion of particle deviations is made around the bilayer positions, with a free lattice parameter determined from a variational approach. For each of the two walls, the harmonic expansion implies an effective one-body potential at the root of all observables of interest in our Wigner Strong-Coupling expansion. Analytical results for the particle density profile and the pressure are…
We probe the electrostatic mechanism driving adsorption of polyelectrolytes onto like-charged membranes upon the addition of tri- and tetravalent counterions to a bathing monovalent salt solution. We develop a one-loop-dressed strong coupling theory that treats the monovalent salt at the electrostatic one-loop level and the multivalent counterions within a strong-coupling approach. It is shown that the adhesive force of the multivalent counterions mediating the like-charge adsorption arises from their strong condensation at the charged membrane. The resulting interfacial counterion excess loca…
We consider an overall neutral system consisting of two similarly charged plates and their oppositely charged counterions and analyze the electrostatic interaction between the two surfaces beyond the mean-field Poisson-Boltzmann approximation. Our physical picture is based on the fluctuation-driven counterion condensation model, in which a fraction of the counterions is allowed to ``condense'' onto the charged plates. In addition, an expression for the pressure is derived, which includes fluctuation contributions of the whole system. We find that for sufficiently high surface charges, the dist…
We predict a condensation phenomenon in an overall neutral system, consisting of a single charged plate and its oppositely charged counterions. Based on the ``two-fluid'' model, in which the counterions are divided into a ``free'' and a ``condensed'' fraction, we argue that for high surface charge, fluctuations can lead to a phase transition in which a large fraction of counterions is condensed. Furthermore, we show that depending on the valence, the condensation is either a first-order or a smooth transition.
We investigate end-effects in the ion distribution around strongly charged, flexible polyelectrolytes with a quenched charge distribution by molecular dynamics simulations of dilute polyelectrolyte solutions. We take the counterions explicitly into account and calculate the full Coulomb interaction via an Ewald summation method. We find that the free counterions of the solution are distributed in such a way that a fraction of the chain charges is effectively neutralized. This in turn leads to an effective charge distribution which is similar to those found for weakly charged titrating polyelec…
Surface sensitive synchrotron-X-ray scattering studies reveal the distributions of monovalent ions next to highly charged interfaces. A lipid phosphate (dihexadecyl hydrogen-phosphate) was spread as a monolayer at the air-water interface, containing CsI at various concentrations. Using anomalous reflectivity off and at the $L_3$ Cs$^+$ resonance, we provide, for the first time, spatial counterion distributions (Cs$^+$) next to the negatively charged interface over a wide range of ionic concentrations. We argue that at low salt concentrations and for pure water the enhanced concentration of hyd…
We report results from a molecular dynamics (MD) simulation on the conformations of a long flexible polyelectrolyte complexed to a charged sphere, \textit{both negatively charged}, in the presence of neutralizing counterions in the strong Coulomb coupling regime. The structure of this complex is very sensitive to the charge density of the polyelectrolyte. For a fully charged polyelectrolyte the polymer forms a dense two-dimensional "disk", whereas for a partially charged polyelectrolyte the monomers are spread over the colloidal surface. A mechanism involving the \textit{overcharging} of the p…
We consider the Dean-Kawasaki (DK) equation of overdamped Brownian particles that forms the basis of the stochastic density functional theory. Recently, the linearized DK equation has successfully reproduced the full Onsager theory of symmetric electrolyte conductivity. In this paper, the linear DK equation is applied to investigate density fluctuations around the ground state distribution of strongly coupled counterions near a charged plate, focusing especially on the transverse dynamics along the plate surface. Consequently, we find a crossover scale above which the transverse density dynami…
Light scattering and viscometric measurements on weak polyelectrolytes show two important aspects of counterion condensation, namely, non-monotonic variation in the polyelectrolyte size with the increase in the electrostatic strength, and, monovalent counterion selectivity in determining the nature of collapse transition at high electrostatic strengths. Here, we present a self-consistent variational theory for weak polyelectrolytes which includes the effects of the polarizability of monovalent counterions. Our theory reproduces several experimental findings including non-monotonic conformation…
We theoretically studied electric double layer by using the mean-field approach including the non-uniform size effect and the orientational ordering of water dipoles in electrolyte solution. Performing the minimization of the free energy, the resulting ion and water distribution functions are determined numerically in the process of solving the Poisson's equation. The effect of non-uniform ionic size and orientational ordering of water dipoles on stratification of counterions of different size in multicomponent electrolyte solution near the charged surface are presented and discussed. Spatial …
We use coarse-grained Monte Carlo simulations to study the elastic properties of charged membranes in solutions of monovalent and pentavalent counterions. The simulation results of the two cases reveal trends opposite to each other. The bending rigidity and projected area increase with the membrane charge density for monovalent counterions, while they decrease for the pentavalent ions. These observations can be related to the counterion screening of the lipid charges. While the monovalent counterions only weakly screen the Coulomb interactions, which implies a repulsive Coulomb system, the mul…
We review works on screening of a macroion, such as colloidal particle or double helix DNA, by multivalent counterions. Multivalent metal ions, charged micelles, short or long polyelectrolytes can play the role of multivalent counterions. Due to the strong Coulomb repulsion such multivalent counterions form a strongly correlated liquid resembling a Wigner crystal at the surface of the macroion. Even if the macroion is neutralized by this liquid, a newly arriving counterion creates in the liquid a correlation hole or image which attracts the ion to the surface. As a result, total charge of adso…
We perform extensive molecular dynamics simulations of a highly charged flexible polyelectrolyte (PE) chain in a poor solvent for the case when the chain is in a collapsed state and the electrostatic interactions, characterized by the reduced Bjerrum length $\ell_B$, are strong. We detect the existence of several sub-regimes, $R_g \sim \ell_B^{-γ}$, in the dependence of the gyration radius of the chain $R_g$ on $\ell_B$. In contrast to a good solvent, the exponent $γ$ for a poor solvent crucially depends on the size and valency of counterions. To explain the different sub-regimes we generalize…
In one and two component solvents, we calculate the counterion distribution around a charged rod treating the degree of ionization $α$ as an annealed variable dependent on its local environment. In the one component case, $α$ is determined under various conditions without and with salt. In the two component case, we take into account the preferential solvation of the counterions and the ionized monomers and the short-range interaction between the rod and the solvent without salt. It then follows a composition-dependent mass action law. Mesoscopic variations of the composition and the counterio…
We investigate electrostatic stability of charged droplets, modeled as permeable, charged spheres, and their encapsidation in thin, arbitrarily charged nano-shells, immersed in a neutralizing asymmetric electrolyte background. The latter consists of a small concentration of mobile multivalent counterions in a bathing solution of monovalent (positive and negative) ions. We use extensive Monte-Carlo simulations to investigate the spatial distribution of multivalent counterions and the electrostatic component of their osmotic pressure on the bounding surface of the spherical nano-shell. The osmot…
Screening of a strongly charged macroion by its multivalent counterions can not be described in the framework of mean field Poisson-Boltzmann (PB) theory because multivalent counterions form a strongly correlated liquid (SCL) on the surface of the macroion. It was predicted that a distant counterion polarizes the SCL as if it were a metallic surface and creates an electrostatic image. The attractive potential energy of the image is the reason why the charge density of counterions decreases faster with distance from the charged surface than in PB theory. Using the Monte Carlo method to find the…
We formulate an extension of the Debye-Hueckel-Bjerrum theory [M. E. Fisher and Y. Levin, Phys. Rev. Lett. 71, 3826 (1993)] to the fluid state of a highly asymmetric charged colloid. Allowing for the formation of clusters consisting of one polyion and n condensed counterions, the total Helmholtz free energy of the colloidal suspension is constructed. The thermodynamic properties, such as the cluster-density distribution and the pressure, are obtained by the minimization of the free energy under the constraints of fixed number of polyions and counterions. In agreement with the current experimen…
We explore, using the recently developed efficient Monte Carlo simulation method, the interaction of an anionic polyelectrolyte solution with a like-charged dielectric surface. In addition to polyions, the solution also contains salt with either monovalent, divalent, or trivalent counterions. In agreement with recent experimental observations, we find that multivalent counterions can lead to strong adsorption of polyions onto the surface. On the other hand, addition of a 1:1 electrolyte diminishes the adsorption induced by the multivalent counterions. Dielectric discontinuity at the interface …
Screening of a strongly charged macroion by oppositely charged colloidal particles, micelles, or short polyelectrolytes is considered. Due to strong lateral repulsion such multivalent counterions form a strongly correlated liquid at the surface of the macroion. This liquid provides correlation induced attraction of multivalent counterions to the macroion surface. As a result even a moderate concentration of multivalent counterions in the solution inverts the sign of the net macroion charge. We show that at high concentration of monovalent salt the absolute value of inverted charge can be large…
We study the distribution of multivalent counterions next to a dielectric slab, bearing a quenched, random distribution of charges on one of its solution interfaces, with a given mean and variance, both in the absence and in the presence of a bathing monovalent salt solution. We use the previously derived approach based on the dressed multivalent-ion theory that combines aspects of the strong and weak coupling of multivalent and monovalent ions in a single framework. The presence of quenched charge disorder on the charged surface of the dielectric slab is shown to substantially increase the de…
Soft matter materials, such as polymers, membranes, proteins, are often electrically charged. This makes them water soluble, which is of great importance in technological application and a prerequisite for biological function. We discuss a few static and dynamic systems that are dominated by charge effects. One class comprises complexation between oppositely charged objects, for example the adsorption of charged ions or charged polymers (such as DNA) on oppositely charged substrates of different geometry. The second class comprises effective interactions between similarly charged objects. Here…
In order to detect the effect of the surface charge discreteness on the properties at the solid-liquid interface, molecular dynamics simulation model taking consideration of the vibration of wall atoms was used to investigate the ion and water performance under different charge distributions. Through the comparison between simulation results and the theoretical prediction, it was found that, with the degree of discreteness increasing, much more counterions were attracted to the surface. These ions formed a denser accumulating layer which located much nearer to the surface and caused charge inv…
By decomposing the Coulomb interaction into a long distance component appropriate for mean-field theory, and a nonmean-field short distance component, we compute the counterion density near a charged surface for all values of the counterion coupling parameter. A modified strong-coupling expansion that is manifestly finite at all coupling strengths is used to treat the short distance component. We find a nonperturbative correction related to the lateral counterion correlations that modifies the density at intermediate coupling.
A highly-charged spherical colloid in a salt-free environment exerts such a powerful attraction on its counterions that a certain fraction condenses onto the surface of a particle. The degree of condensation depends on the curvature of the surface. So, for instance, condensation is triggered on a highly-charged sphere only if the radius exceeds a certain critical radius $\collrad^{*}$. $\collrad^{*}$ is expected to be a simple function of the volume fraction of particles. To test these predictions, we prepare spherical particles which contain a covalently-bound ionic liquid, which is engineere…
Using a field-theoretic approach, we derive the first few coefficients of the exact low-density (``virial'') expansion of a binary mixture of positively and negatively charged hard spheres (two-component hard-core plasma, TCPHC). Our calculations are nonperturbative with respect to the diameters $d_+$ and $d_-$ and charge valences $q_+$ and $q_-$ of positive and negative ions. Consequently, our closed-form expressions for the coefficients of the free energy and activity can be used to treat dilute salt solutions, where typically $d_+ \sim d_-$ and $q_+ \sim q_-$, as well as colloidal suspensio…
The effective elasticity of highly charged stiff polyelectrolytes is studied in the presence of counterions, with and without added salt. The rigid polymer conformations may become unstable due to an effective attraction induced by counterion density fluctuations. Instabilities at the longest, or intermediate length scales may signal collapse to globule, or necklace states, respectively. In the presence of added-salt, a generalized electrostatic persistence length is obtained, which has a nontrivial dependence on the Debye screening length.
We study a two-dimensional model for a long cylindrical stiff charged macroion immersed in a charge-asymmetric electrolyte with charge ratio +2/-1. The model is integrable and it allows an exact analytical determination of the effective charge of the macroion, which characterizes the electrostatic potential at large distances (compared to the screening length) from the macroion. At at high coulombic coupling, this model predicts charge inversion: for a highly negatively charged macroion, the effective charge could become positive, indicating an overscreening of the macroion by the divalent cou…
